Central extensions of loop groups and obstruction to pre-quantization

TL;DR

This paper constructs a pre-quantum line bundle for the moduli space of flat G-bundles over a Riemann surface, clarifying the relationship between central extensions of loop groups and pre-quantization obstructions.

Contribution

It provides an explicit construction of a pre-quantum line bundle for non-simply connected compact simple Lie groups, linking loop group extensions to pre-quantization obstructions.

Findings

Explicit pre-quantum line bundle construction for non-simply connected groups

Clarification of the relationship between loop group extensions and pre-quantization obstructions

Explanation of the coincidence observed in prior classifications

Abstract

An explicit construction of a pre-quantum line bundle for the moduli space of flat G-bundles over a Riemann surface is given, where G is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano-Laredo's work classifying central extensions of loop groups LG and the author's previous work on the obstruction to pre-quantization of the moduli space of flat G-bundles.